Reduction Puzzles

I admit that I’m addicted to patterns and tessellations. This blog is my attempt to catalog all the patterns generated from a single unit. But, I’m not content to leave it at that.

Each pattern shown here can generate new patterns. How? It’s simple: identify the shapes in a pattern, then combine them into new shapes.

I call these “Reduction Puzzles” because combining the shapes reduces the number of shapes in the tessellation. A 5-shape tessellation can become a 4-shape tessellation can become a 3-shape tessellation can become a 2-shape tessellation can become a 1-shape tessellation.

Let me show you an example.

Here’s Pattern 1-1-1-3M/p2/III:

This pattern is a 5-shape tessellation. Below, I’ve shaded the pattern in black, white, and gray.

Here are each of the five shapes, lettered A through E.

Shape A comes in contact with itself, so it requires two colors: I happen to use black and white.

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Now, I’ll systematically combine shape A with shapes B, C, D, and E.

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Shape B only comes in contact with shape A or shape D.

Combining these two shapes produces two different shapes because not all Bs touch all Ds.

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Shape C only comes in contact with shape A or shape E.

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Do you see where this is headed? Combining two shapes in the examples above has changed this from a 5-shape tessellation to a 4-shape tessellation. In a similar manner, combining three shapes will change it from a 5-shape tessellation to a 3-shape tessellation; combining four shapes will change it to a 2-shape tessellation; and, combining all five shapes will change it to a 1-shape tessellation, in this case, a sorta wavy stripe.

Other patterns will give different results.  Try out several for yourself.